Polynomial Approximation on Polytopes
Polynomial Approximation on Polytopes

Author: Vilmos Totik

Publisher: American Mathematical Soc.

Published: 2014-09-29

Total Pages: 124

ISBN-13: 1470416662

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Polynomial approximation on convex polytopes in is considered in uniform and -norms. For an appropriate modulus of smoothness matching direct and converse estimates are proven. In the -case so called strong direct and converse results are also verified. The equivalence of the moduli of smoothness with an appropriate -functional follows as a consequence. The results solve a problem that was left open since the mid 1980s when some of the present findings were established for special, so-called simple polytopes.

Positive Polynomials, Convex Integral Polytopes, and a Random Walk Problem
Positive Polynomials, Convex Integral Polytopes, and a Random Walk Problem

Author: David Handelman

Publisher: Springer

Published: 1987

Total Pages: 156

ISBN-13:

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Emanating from the theory of C*-algebras and actions of tori theoren, the problems discussed here are outgrowths of random walk problems on lattices. An AGL (d,Z)-invariant (which is a partially ordered commutative algebra) is obtained for lattice polytopes (compact convex polytopes in Euclidean space whose vertices lie in Zd), and certain algebraic properties of the algebra are related to geometric properties of the polytope. There are also strong connections with convex analysis, Choquet theory, and reflection groups. This book serves as both an introduction to and a research monograph on the many interconnections between these topics, that arise out of questions of the following type: Let f be a (Laurent) polynomial in several real variables, and let P be a (Laurent) polynomial with only positive coefficients; decide under what circumstances there exists an integer n such that Pnf itself also has only positive coefficients. It is intended to reach and be of interest to a general mathematical audience as well as specialists in the areas mentioned.

Polytopes
Polytopes

Author: Tibor Bisztriczky

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 515

ISBN-13: 9401109249

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The aim of this volume is to reinforce the interaction between the three main branches (abstract, convex and computational) of the theory of polytopes. The articles include contributions from many of the leading experts in the field, and their topics of concern are expositions of recent results and in-depth analyses of the development (past and future) of the subject. The subject matter of the book ranges from algorithms for assignment and transportation problems to the introduction of a geometric theory of polyhedra which need not be convex. With polytopes as the main topic of interest, there are articles on realizations, classifications, Eulerian posets, polyhedral subdivisions, generalized stress, the Brunn--Minkowski theory, asymptotic approximations and the computation of volumes and mixed volumes. For researchers in applied and computational convexity, convex geometry and discrete geometry at the graduate and postgraduate levels.

Introduction To The Theory Of Weighted Polynomial Approximation
Introduction To The Theory Of Weighted Polynomial Approximation

Author: H N Mhaskar

Publisher: World Scientific

Published: 1997-01-04

Total Pages: 398

ISBN-13: 9814518050

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In this book, we have attempted to explain a variety of different techniques and ideas which have contributed to this subject in its course of successive refinements during the last 25 years. There are other books and surveys reviewing the ideas from the perspective of either potential theory or orthogonal polynomials. The main thrust of this book is to introduce the subject from an approximation theory point of view. Thus, the main motivation is to study analogues of results from classical trigonometric approximation theory, introducing other ideas as needed. It is not our objective to survey the most recent results, but merely to introduce to the readers the thought processes and ideas as they are developed.This book is intended to be self-contained, although the reader is expected to be familiar with rudimentary real and complex analysis. It will also help to have studied elementary trigonometric approximation theory, and have some exposure to orthogonal polynomials.

Approximation by Polynomials with Integral Coefficients
Approximation by Polynomials with Integral Coefficients

Author: Le Baron O. Ferguson

Publisher: American Mathematical Soc.

Published: 1980

Total Pages: 174

ISBN-13: 0821815172

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Addresses two questions that include: 'What functions can be approximated by polynomials whose coefficients are integers?' and 'How well are they approximated (Jackson type theorems)?'

Theory of Uniform Approximation of Functions by Polynomials
Theory of Uniform Approximation of Functions by Polynomials

Author: Vladislav K. Dzyadyk

Publisher: Walter de Gruyter

Published: 2008-09-25

Total Pages: 497

ISBN-13: 3110208245

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A thorough, self-contained and easily accessible treatment of the theory on the polynomial best approximation of functions with respect to maximum norms. The topics include Chebychev theory, Weierstraß theorems, smoothness of functions, and continuation of functions.

Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Topics in Hyperplane Arrangements, Polytopes and Box-Splines

Author: Corrado De Concini

Publisher: Springer Science & Business Media

Published: 2010-08-18

Total Pages: 387

ISBN-13: 0387789634

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Topics in Hyperplane Arrangements, Polytopes and Box-Splines brings together many areas of research that focus on methods to compute the number of integral points in suitable families or variable polytopes. The topics introduced expand upon differential and difference equations, approximation theory, cohomology, and module theory. This book, written by two distinguished authors, engages a broad audience by proving the a strong foudation. This book may be used in the classroom setting as well as a reference for researchers.